What's New
About the Project
NIST
10 Bessel FunctionsSpherical Bessel Functions

§10.60 Sums

Contents

§10.60(i) Addition Theorems

Define u, v, w, and α as in §10.23(ii). Then with Pn again denoting the Legendre polynomial of degree n,

10.60.1 cosww=-n=0(2n+1)jn(v)yn(u)Pn(cosα),
|ve±iα|<|u|.
10.60.2 sinww=n=0(2n+1)jn(v)jn(u)Pn(cosα).
10.60.3 e-ww=2πn=0(2n+1)in(1)(v)kn(u)Pn(cosα),
|ve±iα|<|u|.

§10.60(ii) Duplication Formulas

10.60.4 jn(2z)=-n!zn+1k=0n2n-2k+1k!(2n-k+1)!jn-k(z)yn-k(z),
10.60.5 yn(2z)=n!zn+1k=0nn-k+12k!(2n-k+1)!(jn-k2(z)-yn-k2(z)),
10.60.6 kn(2z)=1πn!zn+1k=0n(-1)k2n-2k+1k!(2n-k+1)!kn-k2(z).

§10.60(iii) Other Series

10.60.7 eizcosα =n=0(2n+1)injn(z)Pn(cosα),
10.60.8 ezcosα =n=0(2n+1)in(1)(z)Pn(cosα),
10.60.9 e-zcosα=n=0(-1)n(2n+1)in(1)(z)Pn(cosα).
10.60.10 J0(zsinα)=n=0(4n+1)(2n)!22n(n!)2j2n(z)P2n(cosα).
10.60.11 n=0jn2(z)=Si(2z)2z.

For Si see §6.2(ii).

10.60.12 n=0(2n+1)jn2(z) =1,
10.60.13 n=0(-1)n(2n+1)jn2(z) =sin(2z)2z,
10.60.14 n=0(2n+1)(jn(z))2 =13.

For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000).

§10.60(iv) Compendia

For collections of sums of series relevant to spherical Bessel functions or Bessel functions of half odd integer order see Erdélyi et al. (1953b, pp. 43–45 and 98–105), Gradshteyn and Ryzhik (2000, §§8.51, 8.53), Hansen (1975), Magnus et al. (1966, pp. 106–108 and 123–138), and Prudnikov et al. (1986b, pp. 635–637 and 651–700). See also Watson (1944, Chapters 11 and 16).